Theory of finite-temperature crossovers near quantum critical points close to,or above, their upper-critical dimension

Abstract

A systematic method for the computation of finite-temperature (T) crossover functions near quantum-critical points close to, or above, their upper-critical dimension is devised. We describe the physics of the various regions in the T and critical tuning parameter (t) plane. The quantum-critical point is at T=0, t=0, and in many cases there is a line of finite-temperature transitions at T=${\mathrm{T}}_{\mathrm{c}}$(t), t0, with ${\mathrm{T}}_{\mathrm{c}}$(0)=0. For the relativistic, n-component ${\mathrm{\ensuremath{\varphi}}}^{4}$ continuum quantum field theory [which describes lattice quantum rotor (n\ensuremath{\geqslant}2) and transverse field Ising (n=1) models] the upper-critical dimension is d=3, and for d3, \ensuremath{\epsilon}=3-d is the control parameter over the entire phase diagram. In the region |T-${\mathrm{T}}_{\mathrm{c}}$(t)|\ensuremath{\ll}${\mathrm{T}}_{\mathrm{c}}$(t), we obtain an \ensuremath{\epsilon} expansion for coupling constants which then are input as arguments of known classical, tricritical, crossover functions. In the high-T region of the continuum theory, an expansion in integer powers of $\sqrt{\ensuremath{\epsilon}}$, modulo powers of ln \ensuremath{\epsilon}, holds for all thermodynamic observables, static correlators, and dynamic properties at all Matsubara frequencies; for the imaginary part of correlators at real frequencies (\ensuremath{\omega}), the perturbative $\sqrt{\ensuremath{\epsilon}}$ expansion describes quantum relaxation at \ensuremath{\hbar}\ensuremath{\omega}\ensuremath{\sim}${\mathrm{k}}_{\mathrm{B}}$T or larger, but fails for \ensuremath{\hbar}\ensuremath{\omega}\ensuremath{\sim}$\sqrt{\ensuremath{\epsilon}}$${\mathrm{k}}_{\mathrm{B}}$T or smaller. An important principle, underlying the whole calculation, is the analyticity of all observables as functions of t at t=0, for Tg0; indeed, analytic continuation in t is used to obtain results in a portion of the phase diagram. Our method also applies to a large class of other quantum-critical points and their associated continuum quantum field theories.